\input epsf
\nopagenumbers
\magnification=\magstep1

\noindent {\bf Problem statement} Suppose that $\bf a$ and $\bf b$ are
two three-dimensional vectors.

\medskip

\noindent a) Use the identity $(\cos\theta)^2+(\sin\theta)^2=1$ to
derive an identity involving ${\bf a}\cdot{\bf b}$, ${\bf a}\times{\bf
b}$, $\|{\bf a}\|$, and $\|{\bf b}\|$.

\medskip

\noindent b) Verify this identity algebraically using the formulas
$${\bf a}\cdot{\bf b}=a_1b_1+a_2b_2+a_3b_3; \quad {\bf a}\times{\bf b}=\det\!\pmatrix{{\bf i}&{\bf j}&{\bf k}\cr a_1&a_2&a_3\cr b_1&b_2&b_3\cr}.$$


\vfil\eject\end